$H$-percolation with a random $H$

In $H$-percolation, we start with an Erd\H{o}s--R\'enyi graph ${\mathcal G}_{n,p}$ and then iteratively add edges that complete copies of $H$. The process percolates if all edges missing from ${\mathcal G}_{n,p}$ are eventually added. We find the critical threshold $p_c$ when $H={\mathcal G}_{k,1/2}$ is uniformly random, solving a problem of Balogh, Bollob\'as and Morris.